• A population is the unit about which we want to use statistics to describe, analyse, or perform inference.
  • e.g., all the trees in a forest

• A population is the unit about which we want to use statistics to describe, analyse, or perform inference.
  • e.g., all the trees in a forest
• A sample is a subset of a population that is observed with the goal of obtaining knowledge about a population.

• A population is the unit about which we want to use statistics to describe, analyse, or perform inference.
  • e.g., all the trees in a forest
• A sample is a subset of a population that is observed with the goal of obtaining knowledge about a population.
  • select 7 trees at random
  • Samples must be taken carefully so that they are representative of the entire population!
  • Gold standard: sample sizes must be large and drawn randomly from the population
  • Experimental design: designing the sample such that it represents the desired population

• We can visualise the distribution of values within a population using a histogram

• The workhorse function is hist

# generate some random numbers from the lognormal distribution
my_var = rlnorm(1000, log(10), 0.5)
hist(my_var)

• The workhorse function is hist

The defaults are not very nice, so lets improve things

• main = "" disables the title
• xlab and ylab control axis labels
• breaks controls the number of bins in the histogram
• col sets the color of the bars
• border sets the color of the borders (NA: no border)

hist(my_var, main = "", xlab="Variable of interest", ylab = "Frequency", 
     breaks = 20, col="#999999", border=NA)

• ggplot2 is a package: packages extend the base functionality of R
• You can install packages with install.packages() (just once)
• You load packages with library (once per session)

Using ggplot

• ggplot always works on data.frames, so make sure your data is in one
• Each plot defines aesthetics using aes
  • What to plot
  • In what position (x-axis, y-axis, etc)
  • How to visualise structure in data (e.g., color by categories or variables)
• You then add geometries to tell ggplot how to plot the data
• Here we create a plot, define a single variable plotted in the x position, and set a histogram geometry
• We can easily add other customisations (e.g., axis labels)

• We can visualise the distribution of values within a population using a histogram

• We use summary statistics to describe the values numerically.
• In practise, we must estimate population statistics via sample statistics.
• We can formally describe the shape of the distribution of values using special statistics called moments.

• First moment: Arithmetic mean.

The population mean (\(\mu\)) can be approximated with the sample mean:

\[ \mu \approx \bar{x} = \frac{1}{n}\sum_{i=1}^n x_i \]

• First moment: Arithmetic mean.

Other location statistics

The mean can be strongly influenced by outliers.

• median: The 50% quantile (i.e., half the values of x are above the median, half below)

• First moment: Arithmetic mean.

Other location statistics

The mean can be strongly influenced by outliers.

• median: The 50% quantile (i.e., half the values of x are above the median, half below)
• mode: the most common value, the “peak” of a distribution

• First moment: Arithmetic mean.

Other location statistics

• minimum, maximum values
• quantiles: the p-th quantile of x is the value of x such that p% of values are beneath the quantile
  • commonly used: (2.5%, 97.5%), 50%, (25%, 75%)

Comparing variables

We can compare variables in a way that is location independent by centering (subtracting the mean)

• Second moment: Variance (\(\sigma^2\))

\[ \sigma^2 = \frac{1}{N}\sum_{i=1}^N (X_i-\mu)^2 \]

We can estimate \(\sigma^2\) using the sample variance:

\[ \sigma^2 \approx s^2 = \frac{1}{n-1}\sum_{i=1}^n (x_i -\bar{x})^2 \]

It is convenient to talk about the scale of \(x\) in the same units as \(x\) itself, so we use the (population or sample) standard deviation:

\[ \sigma = \sqrt{\sigma^2} \approx s = \sqrt{s^2} \]

Other dispersion statistics

• range: max(x) - min(x)

• interquartile range (IQR): the difference between the third and first quartiles

• coefficient of variation (CV): \[ \frac{s}{|\bar{x}|} \]

• A boxplot is a nice way to summarize location and dispersion in a dataset

Asymmetry: skewness

Is the distribution weighted to one side or the other?

\[ \mathrm{skewness} = \frac{\sum_{i=1}^{n}(x_i-\bar{x})^3}{(n-1)s^3} \]

• Boxplots (sometimes box-and-whisker diagrams) summarize key statistics.
  • median
  • first and third quartiles (hinges)
  • approx. 95% confidence interval for median (notch)
  • min/max (or quartile + 1.5*IQR) (whiskers)
• They are very useful for comparing variables among groups.
• Base graphics: we use the boxplot function
• y ~ group is a special data type called a formula

• Boxplots (sometimes box-and-whisker diagrams) summarize key statistics.
  • median
  • first and third quartiles (hinges)
  • approx. 95% confidence interval for median (notch)
  • min/max (or quartile + 1.5*IQR) (whiskers)
• They are very useful for comparing variables among groups.
• Base graphics: we use the boxplot function
• y ~ group is a special data type called a formula

• An important programming concept is DRY: Don’t Repeat Yourself!
  • Don’t copy and paste code if you can instead instruct the computer to repeat things for you.
• There are many ways to execute the same code repeatedly; we will cover these in detail as they are needed.
• A motivating example:
  • Compute the mean of every variable in the iris dataset
  • Note that the last variable, species, is nominal and thus the mean is not defined

• There are three species of iris; it makes more sense to have summary statistics for each one
• tapply: tabular apply
  • produces a summary table for a single variable (e.g., Sepal.Length) based on categories in another variable (e.g., Species)

Thought experiment

You and 99 of your closest friends gather on the centre line of a 100-m football pitch. We define the centre line as 0m, the western boundary -50 m, and the eastern boundary as +50 m.

Thought experiment

You and 99 of your closest friends gather on the centre line of a 100-m football pitch. We define the centre line as 0m, the western boundary -50 m, and the eastern boundary as +50 m.

Each person flips a fair coin. If the coin is heads, they take a step east (add 0.5 m to their location), if its tails, they take a step west (subtract 0.5 m from their location).

Question: What is the long-run distribution of positions on the field?

Thought experiment

You and 99 of your closest friends gather on the centre line of a 100-m football pitch. We define the centre line as 0m, the western boundary -50 m, and the eastern boundary as +50 m.

Each person flips a fair coin. If the coin is heads, they take a step east (add 0.5 m to their location), if its tails, they take a step west (subtract 0.5 m from their location).

Question: What is the long-run distribution of positions on the field?

t = 10

Thought experiment

You and 99 of your closest friends gather on the centre line of a 100-m football pitch. We define the centre line as 0m, the western boundary -50 m, and the eastern boundary as +50 m.

Each person flips a fair coin. If the coin is heads, they take a step east (add 0.5 m to their location), if its tails, they take a step west (subtract 0.5 m from their location).

Question: What is the long-run distribution of positions on the field?

t = 50

Thought experiment

You and 99 of your closest friends gather on the centre line of a 100-m football pitch. We define the centre line as 0m, the western boundary -50 m, and the eastern boundary as +50 m.

Each person flips a fair coin. If the coin is heads, they take a step east (add 0.5 m to their location), if its tails, they take a step west (subtract 0.5 m from their location).

Question: What is the long-run distribution of positions on the field?

t = 500

We can simulate this experiment in R.

And for a larger sample (2500) friends, doing more coin flips (500 each), with more concise code.

# repeat the initial position 2500 times, one per friend
n_friends = 2500
positions = rep(0, n_friends)

# repeat the experiment 500 times
time_steps = 1:500
for(i in time_steps) {
    # flip a coin for each friend; this returns one heads or tails for each n_friends
    coin_flips = sample(c("heads", "tails"), size = n_friends, replace = TRUE)
    # compute the steps: +0.5 if the flip is heads, -0.5 if tails
    moves = ifelse(coin_flips == "heads", 0.5, -0.5)
    # finally add the moves to the old positions and save the new position
    positions = positions + moves
}

• We can rescale this histogram, such that the area of each bar represents the proportion of samples within that bin.

• The height of each bar is the probability density.
• The figure approximates an empirical probability density function.

• We can rescale this histogram, such that the area of each bar represents the proportion of samples within that bin.
• The height of each bar is the probability density.
• The figure approximates an empirical probability density function.
• We can use the density function in R to add a curve approximating this density.

• This looks a lot like a normal distribution.
• As we take smaller and smaller steps, and do more coin flips, the distribution of values will converge exactly on a normal distribution.
• Here we add a normal PDF based on the sample mean and sd (in blue).

• The typical “bell-curve” shape of the normal distribution is the Normal Probability Density Function (PDF)

\[ \mathcal{f}(x) = \frac{1}{\sigma \sqrt{2\pi}} \mathcal{e}^{-\frac{1}{2} \left (\frac{x-\mu}{\sigma} \right )^2} \]

• It is easiest to understand what this function represents by understanding its integral.
• Recall that the area of each bar in our histogram represents the proportion of observations in that bin.

• The integral between two values \(a\) and \(b\) is equal to the area under the curve.
• This is exactly the probability of observing a value between \(a\) and \(b\).
• Rule of thumb: about 68% of values in any normal distribution will be within 1 standard deviation of the mean.

• The integral between two values \(a\) and \(b\) is equal to the area under the curve.
• This is exactly the probability of observing a value between \(a\) and \(b\).
• Rule of thumb: about 68% of values in any normal distribution will be within 1 standard deviation of the mean.
• Rule of thumb: 95% of values are \(\mu \pm 1.96 \sigma \approx \mu \pm 2 \sigma\).

• If we integrate from \(-\infty\) to some value \(x\), then we have the probability of observing a value less than \(x\).
• This integral of the Normal PDF is known as the Normal cumulative distribution function.

\[ \mathcal{g}(x) = \int_{-\infty}^{x} \frac{1}{\sigma \sqrt{2\pi}} \mathcal{e}^{-\frac{1}{2} \left (\frac{x-\mu}{\sigma} \right )^2} dx \]